Conventional film bulk acoustic resonators (FBAR) are constructed in the configurations shown in FIGS. 1 and 2. In FIG. 1, a metal electrode 1 is disposed on a layer of piezoelectric material 3 having a thickness T.sub.p, which in turn is disposed on a thinned section of a semiconductor material 5 having a thickness T.sub.s in the thinned area. A second metal electrode 7 is disposed on the semiconductor material, opposed the piezoelectric material 3. In FIG. 2, the second metal electrode 17 is positioned directly between the piezoelectric material 13 having thickness T.sub.p and the thinned semiconductor material 15 having thickness T.sub.s. The first metal electrode 11 is again disposed on the top of the piezoelectric material 13. In both Figures, a voltage applied across the metal electrodes will produce a thickness directed electric field within the piezoelectric material. The bulk acoustic wave generated within the piezoelectric material is usually compressional, since it is easy to grow C-axis oriented zinc oxide (ZnO) or aluminum nitride (AlN) films as the piezoelectric material. A figure-of-merit (FOM) which defines the usefulness of the resonator for band pass filter or voltage controlled oscillator applications is defined as ##EQU1## Q is the unloaded quality factor, and the capacity ratio C/C.sub.o is approximated by ##EQU2## where k.sub.eff.sup.2 is the effective electromechanical coupling constant of the resonator structure, and N is the number of acoustic half wavelengths in the piezoelectric layer. For a well designed, high frequency, energy trapped resonator, the Q is limited by acoustic attenuation in the piezoelectric layer and the semiconducting layer.
If the two configurations shown in FIGS. 1 and 2 are made of the same materials, the Q will be the same for the two devices. There is, however, a difference in the FOM between the two configurations. This difference comes about because the k.sub.eff.sup.2 factor of the two is different. It is easy to show that EQU k.sub.eff.sup.2 =k.sup.2 /{1+(.epsilon..sub.p T.sub.s /.epsilon..sub.s T.sub.p)} (3)
where k.sup.2 is the electromechanical coupling constant of the piezoelectric material, and .epsilon..sub.p, .epsilon..sub.s, and T.sub.p, T.sub.s are the relative dielectric constants and thicknesses of the piezoelectric layer and the substrate, respectively. In equation (3), T.sub.p and T.sub.s have meaning only where the electric field exists, and therefore, for the configuration of FIG. 2, T.sub.s =0. Since k.sub.eff.sup.2 is larger for FIG. 2 than for FIG. 1, it is the preferred configuration for conventional high FOM resonators. It should be noted the efficiency of the resonators shown in both FIGS. 1 and 2 are also affected by acoustic loss in the metal electrodes 1, 7, 11, and 17.
Conventional multi-pole FBAR filters use either the ladder or stacked configuration. The ladder filter shown in FIG. 3 has two or more resonators electrically coupled to each other. In FIG. 3, the semiconductor substrate 31 has two resonators formed in the manner of the resonator of FIG. 2. In FIG. 3, each resonator has piezoelectric material 33 and 33' disposed between metal electrodes 34, 35 and 34', 35' respectively. Input is provided to electrode 34 while output is obtained at electrode 35'. The two resonators are connected by electrically connecting electrodes 35 and 34'. The ladder filter has the advantage that all of the resonators can be made identical to one another. Non-identical frequency (anharmonic) spurious responses are rejected by the cascade of resonators. In the stacked filter arrangement, shown in FIG. 4, the two resonators are deposited one over the other on a semiconductor substrate 41 with a common ground plane separating them. In FIG. 4, input electrode 42 and output electrode 44 are separated from common ground electrode 43 by piezoelectric layers 45 and 45'. The advantage gained by introducing the middle ground plane is that the input and the output are not coupled electrically. The clamped electrical capacity of the resonator still exists, but it appears between input and ground and between output and ground, and it does not connect input to output. The coupling between the input and output is an acoustic coupling. The resonator in this two-port configuration provides better filtering action than a one port device because there are no non-resonant elements connecting input and output. Further, the two-port arrangement eliminates the need for inductors to tune out the input-output capacitance. These inductors are typically an order of magnitude larger in area than the FBAR and tend to dominate a monolithic filter chip. However, the stacked filter has three metal electrodes layers which complicate manufacturing and which contribute to the acoustic loss of the filter.